Question

use graphing to find the solution to the equation
$-x + 3 = |3x + 9|$
a -6
b -1.5
c 1.5 and -3
d -1.5 and -6

Explanation:

Step1: Analyze the equation \( -x + 3 = |3x + 9| \)

The equation involves a linear function \( y = -x + 3 \) and an absolute - value function \( y=|3x + 9|\). The absolute - value function \( y = |3x+9|\) can be split into two piece - wise linear functions:

• When \( 3x + 9\geq0\) (i.e., \(x\geq - 3\)), \(y = 3x + 9\).
• When \( 3x + 9\lt0\) (i.e., \(x\lt - 3\)), \(y=- (3x + 9)=-3x - 9\).

Step2: Find the intersection with \(y=-x + 3\) for \(x\geq - 3\)

Set \( -x + 3=3x + 9\).
Add \(x\) to both sides: \(3=4x + 9\).
Subtract 9 from both sides: \(4x=3 - 9=-6\).
Divide both sides by 4: \(x=\frac{-6}{4}=-1.5\). Since \(-1.5\geq - 3\), this is a valid solution.

Step3: Find the intersection with \(y=-x + 3\) for \(x\lt - 3\)

Set \( -x + 3=-3x - 9\).
Add \(3x\) to both sides: \(2x + 3=-9\).
Subtract 3 from both sides: \(2x=-9 - 3=-12\).
Divide both sides by 2: \(x=-6\). Since \(-6\lt - 3\), this is a valid solution.

Answer:

D. - 1.5 and - 6